Precision and Stability Issues in VBL, the Virtual Biophysics Lab simulation program

TL;DR

This paper discusses the development of the Virtual Biophysics Lab (VBL), a simulation program for tumor spheroids that models complex biochemical and diffusion processes, emphasizing the importance of stable numerical integration methods.

Contribution

It highlights the need for unconditionally stable integrators in simulating complex biological systems and reviews mathematical principles to improve simulation accuracy.

Findings

Identification of stability issues in existing simulation methods

Development of integrators that ensure numerical stability

Emphasis on mathematical principles for biological system modeling

Abstract

The network of biochemical reactions inside living organisms is characterized by an overwhelming complexity which stems from the sheer number of reactions and from the complicated topology of biochemical cycles. However the high speed of computers and the sophisticated computational methods that are available today are powerful tools that allow the numerical exploration of these exceedingly interesting dynamical systems. We are now developing a program, the Virtual Biophysics Lab (VBL), that simulates tumor spheroids, and which includes a reduced - but still quite complex - description of the biochemistry of individual cells, plus many diffusion processes that bring oxygen and nutrients into cells and metabolites into the environment. Each simulation step requires the integration of nonlinear differential equations that describe the individual cell's clockwork and the integration of the diffusion equations. These integrations are carried out under widely different conditions, in a changing environment, and for this reason they need integrators that are both unconditionally stable and that do not display unwanted algorithmic artifacts. These conditions are not always fulfilled in the existing literature, and we feel that a review of the underlying mathematical principles may be important not just for us but for other workers in the field of system biology as well.